Vast Portfolio Selection With Submodular Norm Regularizations

Zepeng Zhang, Ziping Zhao

Abstract: Portfolio selection (or portfolio optimization) has been a fundamental problem in the financial investment world since the modern portfolio theory (a.k.a. mean-variance analysis) was introduced by Harry Markowitz in 1952. The goal of portfolio selection is to assign different portions of dollars to the underlying assets according to a certain investment target. In practice, to overcome the high sensitivity to inevitable estimation errors of the input parameters, regularization techniques have been introduced for portfolio stabilization. Besides, regularization techniques have been shown to be capable of achieving some other goals like selecting sparse portfolios and grouping assets whose returns exhibit colinearity. While achieving such merits, acclaimed regularizers like the l <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> -norm may bring an adverse effect on the original portfolio design target. In this paper, a novel framework for vast portfolio selection is proposed via the lens of submodular set functions, which can select a sparse portfolio according to the influence that each asset exerts on the overall portfolio risk. An efficient and convergent algorithm based on the alternating direction method of multipliers is developed for problem resolution. The superiority of the proposed portfolio selection framework is demonstrated with numerical simulations on real market data.